


Case 4

by scatterglory



Category: Stargate Atlantis
Genre: stats!fic
Language: English
Status: Completed
Published: 2010-01-13
Updated: 2010-01-13
Packaged: 2017-10-06 06:02:00
Rating: Mature
Warnings: Creator Chose Not To Use Archive Warnings
Chapters: 1
Words: 2,317
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/50453
Author URL: https://archiveofourown.org/users/scatterglory/pseuds/scatterglory
Summary: <blockquote class="userstuff">
              <p>Carson has a hypothesis that needs testing.</p>
            </blockquote>





	Case 4

**Case 4**

 

No one had died in weeks.

No one had even been seriously injured.

There had been no nano or retroviruses, no strange infections, no mental breakdowns. Aside from another one of Rodney's pretend-I'm-dying-when-it's-really-gas incidents—and who had really known about that side effect of Athosian cooking, anyway?—the infirmary had been . . . dead.

Carson looked at the computer screen and sighed. Data on the wraith, the crewmembers, the various peoples they'd encountered in the Pegasus galaxy. He'd analyzed it hundreds of times. Granted, each time he discovered something new—some minute trend, minor discrepancy—but all the same, he was . . .

Bored.

He sighed again. It should have been absolutely impossible to be bored on Atlantis. Different galaxy, for God's sake! There should be hundreds of thousands of things to occupy his time: patching up teams after dangerous missions, making new discoveries, contributing invaluable information to the field of medicine . . .

Only—not.

The teams came back unscathed, the discoveries had been run-of-the-mill, and all of his recent findings had been mundane at best.

So he'd been left with an unprecedented amount of free time. At first, he'd been utterly at a loss as to how to occupy himself—though flying practice with Major Sheppard was nothing if not educational, he shuddered to think of the long-term damage to his blood pressure—but recently, he'd noticed his attention being drawn again and again to a certain topic . . .

**Introduction to Applied Biostatistics**

Surgery. Genetics. General medicine. Carson was known by everyone on Atlantis to be exceptionally skilled in almost every medical arena. What most of his colleagues did not know, however—he'd almost revealed his aptitude to Rodney when the ever-superior scientist had compared medicine to _voodoo_, but had managed to keep his mouth shut—was that he also harbored a deep-seated passion for the field of Biostatistics. (In an environment as closed as Atlantis, it was nice to have a personal secret . . . and if no one knew, no one could tease him . . . not that he was embarrassed, of course.)

Biostats was a kind of job-related hobby for him—aside from the necessity of possessing a basic understanding of statistics in order to conduct effective medical research, Carson enjoyed the way numbers and statistical concepts could be applied to almost every area of life.

Over the last few days, for example, a hazy hypothesis had been solidifying in his mind. No one event in particular had caught his attention, but he'd been accumulating small observations—inflections, gestures, subtle vibes—that had piqued his curiosity regarding the (statistical, of course) relationship between two of his closest colleagues:

Major John Sheppard and Dr. Rodney McKay.

**Hypothesis Testing Procedure for a Two-Sample Hypothesis**

**1\. Hypothesis Formation**

As every statistician knows, the first step of performing a hypothesis test is to establish two hypotheses:

1\. The null hypothesis, or what is assumed/appears to be true.

2\. The alternative hypothesis, or new theory/information that must be tested.

(Of course, if the null hypothesis is determined to be correct, the alternative hypothesis must be false, and vice versa.)

_Carson leans back in his chair and considers._

Null hypothesis: the relationship between Sheppard and McKay is exactly what everyone on Atlantis assumes it to be—purely professional, overtones of friendship.

Alternative hypothesis: the relationship is . . . not merely what everyone assumes.

**2\. The Significance Level/Acceptable Level of Error**

After determining the hypothesis, the statistician must decide what the appropriate level of error for the test will be; that is, what is the probability that the alternative hypothesis can occur, given that the null hypothesis is correct.

_Carson doesn't see how it's possible for the relationship to be both more than and the same as expected; however, there's still a slim chance (allowing for touchy definitions and semantic obfuscation). He sets the appropriate significance/error level at the traditional 5%._

**3\. Test statistic and Power of the Test**

The next step is to determine the actual test statistic value for the alternative hypothesis, using the—

_Carson kicks himself. He's completely forgotten to determine the case of the problem! This complicates things . . . _

**(2.5 Preliminary Test of Variances)**

An integral part of calculating the test statistic is knowing the variances of the samples. The variances, or degrees to which the samples diverge from the mean (average or norm) can be either equal (Case 2) or unequal (Case 3).

_Lacking concrete data about the specific variances of Sheppard and McKay, Carson can't perform the preliminary F-test of variance; he can, however, use deductive reasoning in the place of data and follow the preliminary test model._

_(Carson disregards the possibility that the test can fall under Case 1. Case 1 refers only to samples with known, quantified variances, and Carson has no idea how to quantify McKay's and Sheppard's deviations from the norm.)_

Null hypothesis for preliminary test: the variances are equal.  
Alternative hypothesis: the variances are not equal.

_Carson deliberates. If he assumes that the variances are equal, he can pool them together and use the standard pooled variance in this calculation of the test statistic (the Case 2 method). However, Carson does not believe that McKay's and Sheppard's variations from the average can be comprehensively combined; therefore, he must assume that they are not equal. Very well. Case 3._

**Back to Step 3.**

The significance level for this test has been set at 5%, or 0.05. This is known as the probability of a Type I Error, or rejecting the null hypothesis although it is true.

_Committing a Type I Error could be very uncomfortable—Carson does not want to assume that there is more passing between Sheppard and McKay than there actually is. However, neither does he want to commit a Type II Error . . . _

A Type II Error entails _not_ rejecting the null hypothesis although it is false. The Power of the Test is the probability that a Type II Error will not be committed.

_Ideally, Carson thinks, he would be able to limit the probabilities of committing both Type I and Type II Errors; however, they are inversely related. This means that the less likely it is that he will inaccurately attribute the properties of a close relationship to Sheppard and McKay, the more likely it is that he will fail to detect the presence of a relationship, should one exist. As a matter of principle, statisticians prefer to minimize the probability of a Type I Error and allow the Type II Error increase; however, Carson believes that for the purpose of this test, it will be more effective to allow his Type Error I (accidentally concluding the presence of a relationship) to increase, which in turn will increase the ability of his test to detect a relationship if present (by decreasing the Type II Error)._

A large sample size is also needed to ensure the Power of the Test. However, if the effect size of the sample is large enough, a smaller sample size may be sufficient.

_Carson decides to collect a small number of samples that have a large effect size to support the high Power of his test. Turning back to his computer, he searches the official reports of all Atlantis missions run by Sheppard and his team. With only a slight pang of conscience, he uses his medical clearance to access the personal reports submitted by Sheppard and McKay regarding each of the missions._

**4\. The Rejection Rule**

Before examining the data, the statistician must establish the conditions under which the null hypothesis may be rejected. By establishing these conditions prior to the evaluation of the data, the statistician avoids biasing the results of the experiment by warping the data to fit the alternative hypothesis.

_Carson decides that this rejection region will be based on the emotional content of the reports. If they contain more than a normal, professional amount of significant emotions, he will reject his null hypothesis. He begins to read._

_* * *_

_Several hours later, Carson has a sample of four relevant reports and a headache._

1\. McKay risks his life to lure a hostile creature of living darkness into the Stargate as Sheppard and everyone else watches helplessly.

2\. Sheppard is attacked by a life-sucking alien bug and McKay must beat the clock to get him back to Atlantis.

3\. Sheppard is saved by the Daedalus at the last possible moment before completing a suicide mission to a hive ship during the siege on Atlantis.

4\. McKay almost kills both of them when his experiment with an Ancient weapon destroys nearly an entire solar system.

_Each report deals with the near-death of one or both of them, and yet even their personal reports express nothing more than what looks like a normal amount of concern, fear and/or anger. Their emotions run slightly higher than a purely professional relationship would suggest, but they still fall well outside the rejection region of friendship._

**5\. Conclusion**

_Carson turns off his computer, disheartened. _

The data does not provide sufficient evidence to suggest that the null hypothesis is false.

_Whatever relationship exists between Sheppard and McKay must be exactly what it appears: two close colleagues, teammates and friends, and nothing more._

_* * *_

_Carson eats dinner that night with Elizabeth, Teyla, Ronon, Radek . . . and John and Rodney. Although tempted to forgo Elizabeth's invitation—seeing John and Rodney would only remind him of his wasted afternoon—his curiosity gets the better of him and he decides to see if he can reconcile his gut feeling with the result of his test._

_Dinner is pleasant enough. Radek and Rodney may have deduced the whereabouts of another ZPM, the Athosians via Teyla report a plentiful harvest, Ronon handed John's squad their collective asses again during a sparring session, and Elizabeth has finalized a mutually-beneficial trade agreement with a new race. Carson eats quietly, still slightly disgruntled, and watches his friends._

_Ronon steals a roll off of Rodney's tray as the scientist banters with Radek; Rodney cries insult and threatens to use Ronon as a bartering chip the next time they encounter a race with something to trade._

_"We'll just offer them one ornery barbarian for their measly ZedPM, and kill two birds with one stone—"_

_Ronon brandishes a lemon-like fruit wordlessly, still munching on Rodney's roll. Rodney recoils in horror and opens his mouth—_

_"Chill, McKay. Beckett's right here in case any of that malicious juice gets into your food," John calmly forestalls the imminent outburst as he replaces Rodney's stolen roll with one of his own._

_"That's not the point," Rodney complains as he transfers the green—something—that he can't stand but John loves onto the pilot's plate. "The point is a complete disregard for my well-being. What if he poisons me someday when Carson's not here? I could go into anaphylactic shock and die, and—"_

_"—and then where would Atlantis be?" finished John, accepting the greens._

_"Exactly," Rodney concludes, satisfied. _

_And then it hits Carson. "Excuse me, I just remembered something I have to do—"_

_He's out the door and down the hall before anyone can protest._

_* * *_

_He drops back into his chair and switches on the computer, grinning like an idiot. No wonder the test came up negative! He'd been using the wrong case the entire time . . .   
_

**Case 4**

The fourth case for hypothesis testing deals with a single sample that changes over time, using paired or matched data.

_Carson had assumed that the relationship between Sheppard and McKay was one of two distinct samples, and therefore either Case 2 or Case 3. He'd completely ignored the final possibility: rather than being two separate samples, McKay and Sheppard represent paired, dependant data. So. New hypothesis._

Null hypothesis: There has been no change in the relationship over time.

Alternative hypothesis: There has been a measurable change in the relationship over time.

_Carson leaves his Significance Level at 0.05 and can now completely ignore the Power of the Test. That leaves calculating the test statistic, or the difference in the relationship over time as affected by the McKay/Sheppard unit's variation from the norm._

_He scans the individual reports again; not surprisingly, they are of no help. Of course—individual considerations have no place in dependant data studies. Smiling to himself, he accesses all the group reports for his four-mission sample—mission preparation, updates, and debriefings._

_There!_

_Carson laughs out loud, delighted. There it is, right in front of him. He turns the audio off of the video recordings, and watches the four reports one by one._

_First report: Sheppard and McKay stand on opposite ends of the briefing room with Teyla, Ford, Elizabeth and himself between them._

_Second report: Sheppard and McKay stand on the same side of the room._

_Third report: Sheppard and McKay stand within arm's length of each other. Judging by the near-simultaneous movement of their mouths, they are finishing each other's sentences._

_Fourth report: Sheppard sits on Elizabeth's desk, arms folded across his chest and completely relaxed. McKay vibrates with energy and paces—but his pacing stays within a small orbit of the Major, often passing by close enough that their uniforms touch. Sheppard's eyes track McKay every second his attention is not actively claimed by whoever is currently speaking._

_Carson grins. A change over time, indeed. And coupled with their behavior tonight . . . _

_His grin widens. To give even more support to the findings, it seems that their body-positioning in the fourth report is a mirror-image of Teyla and Ronon's (and Carson _knows _the two of them are sleeping together—it's a physical impossibility for their judiciously-placed and matching bruises, lacerations and bite marks to be coincidental)._

_He turns his computer off (again) and spins in his chair. _

Conclusion:

_He walks to the door._

The data does provide significant evidence to suggest that the alternative hypothesis is correct.

_Still smiling, he turns off the light and leaves the infirmary._

The sample relationship has indeed changed over time.

_fin_


End file.
